A method for the analysis of stress in brittle rock

A method for the analysis of stress in brittle rock

Int. J. Rock Mech. Min. Sci. Vol. 9, pp. 87-102. Pergamon Press 1972. Printed in Great Britain A METHOD FOR THE ANALYSIS OF STRESS IN BRITTLE ROCK G...

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Int. J. Rock Mech. Min. Sci. Vol. 9, pp. 87-102. Pergamon Press 1972. Printed in Great Britain

A METHOD FOR THE ANALYSIS OF STRESS IN BRITTLE ROCK G. BARLA* Henry Krumb School of Mines, Columbia University, New York

(Received 25 August 1970) Abstraet--A numerical method for the determination of the stresses associated with the fracture processes of a brittle rock material is proposed. It is shown that the various zones of fracture around an underground opening can be predicted and a stability analysis of the rock structure can be performed, while taking the complete physical behavior of the rock material into account. The method is based upon the finite-element analysis and a process of successive approximations. An example of application for a norite rock mass containing a circular opening, under the conditions of plane strain and a uniaxial stress field, is given for illustration. INTRODUCTION FIELD observations reveal that excavations in rock which exhibits a brittle fracture behavior are usually surrounded by a fracture zone. This empirical evidence is supported by the experimental measurement of the velocity of sound in the region around the opening. A definite increase of this velocity with relation to the distance from the opening contour is generally revealed. This indicates that a corresponding increase of the elastic modulus of the rock mass is to be expected. Based on such observations, the rock surrounding the opening may be given different elastic moduli and the boundary of the fracture zone may be located at varying distances fi'om the opening contour. Different assumptions for the geometry and the mechanical properties of the fracture zone may be made [1, 2]. Another approach has been proposed by ZIEN~UEWlCZ, VALLIAPPANand KING [3], who treated the rock as a 'no-tension' material and determined the distribution of stress in several structures. Also in this case, the assumptions introduced are unlikely to represent the nature of the real problem, particularly if structures in rock at great depth are considered. The determination of rock fracture around underground openings is of great importance to the rock mechanics engineer, as safety depends upon the ability to control fracture. The understanding of the re-distribution of stress as fracture progresses is a necessary step in achieving such control. A method, which accounts for the complete physical behavior of a rock material under different loading conditions, contributes to this understanding. It is the purpose of the present paper to propose such a method, which is based upon the use of the finite-element technique and a process of successive approximations. MECHANISM OF BRITTLE FRACTURE The mechanism of brittle fracture of rock as a result of an applied load has been investigated by BIENIAWSKI[4]. Some of his results are used in the course of this paper and are here briefly reviewed. * Present address: Istituto di Arte Mineraria del Politecnico, Torino, Italy. 87

88

G. BARLA

Brittle fracture in multi-axial compression comprises five distinct processes: (1) crack closure, (2) fracture initiation (i.e. stable fracture propagation), (3) critical energy release (i.e. unstable fracture propagation), (4) strength failure and (5) rupture. The same processes occur also in tension. However, crack closure is absent and the processes of stable and unstable fracture propagations virtually non-existent. It is important to notice that the initiation of fracture in compression does not mean that a rock material has lost its load-carrying characteristic. However, in general, a change in the state of the material occurs. Strength failure is a characteristic property of the rock material, while rupture is found to be a fundamental property of the rock structure.

~~//~ ~ U~A~T~RE NSTABLE ........... 7----~"~ ~ PROPAGATION l

El~ FRACTUREINITIATION

CRACKCLOSURE STRAIN FIG. 1. Diagrammatical representation of the complete physical behavior of a brittle rock material (BIENIAWSKI[4]).

The characteristic changes shown in Fig. 1 occur in the mechanical response of a rock material as the applied load is increased. Fracture initiation is manifested by a departure from linearity of the stress-lateral strain curve, with a consequent increase in the value of Poisson's ratio for the rock material. The stress-axial strain curve, however, remains linear, i.e. the modulus of elasticity is constant. The onset of unstable fracture propagation, which is found to coincide with the critical energy release, corresponds to the value of the long-term strength of a rock material. If the load is increased further, the modulus of elasticity decreases while the Poisson's ratio increases due to a considerable cracking which is taking place in the rock. The fracture process is concluded when strength failure is reached. GRIFFITH'S original theory [5] for open cracks and the modification of this theory, due to MCCLI~,rrOCKand WALSH [6] (modified Gritfith criterion), to account for crack closure, have been used for predicting fracture initiation in rock [7]. These studies cannot be used to determine the ultimate stability of a rock structure. However, they are found to be very useful in comparative analyses where the choice is to be made for the best shape and layout of mining excavations [8].

A METHOD FOR THE ANALYSIS OF STRESS IN BRITTLE ROCK

89

For practical purposes, it is possible to assume that the Griffith's theory is valid also to predict the strength failure in tension, as fracture initiation and strength failure in tension occur simultaneously. However, application of the modified Griffith criterion to strength failure in compression is not justified, as has been shown by BIENIAWSKI [9] and BRAC~!, PAULDING and SCHOLZ [10].

METHOD OF ANALYSIS

A rock material under tension reveals a peculiar non-linear behavior pattern of the stressstrain curve, i.e. non-symmetry of the curve with respect to the origin of the coordinate axes (in the following this non-linearity will be referred to as tensile-compressive nonlinearity). For some rock types, this non-linear behavior is quite pronounced, for others it is virtually non-existent. As described in the previous article, non-linear effects are also exhibited by rock during the mechanical response under increasing compressive load. The influence of these different anomalies in the physical behavior of a rock material is neglected in the analyses predicated under the assumption of linear elasticity. The easiest theoretical path by which this simplifying assumption may be removed in any theoretical treatment is to vary the constitutive equations of the rock material. The development of constitutive equations for the description of the various mechanical behavior patterns exhibited by a rock is a subject of intense research. The problem is to establish adequate and practical procedures for the determination of the material functions which enter the various constitutive equations [11]. In the solution of a boundary value problem with non-linear material properties, the use of numerical methods of analysis even for simple geometries, is imperative. In particular, the advantage of the finite-element method in the solution of problems of this type is found in its own nature of formulation, as one can easily treat different constitutive behaviors [12]. The method of analysis presented in this paper consists of solving in a sequence the following separate problems by a process of incremental loading: (a) To determine the distribution of stress in a rock structure where the tensile-compressive non-linearity of the rock material is taken into account. (b) To estimate the zones in which the process of fracture initiation (stable fracture propagation) takes place and to determine the new distribution of stress which follows the inability of the rock to sustain any load in the direction of the tensile stress and the departure from linearity of the stress-lateral strain curve in compression. (c) To establish the onset of unstable fracture propagation in the rock structure and to compute the re-distribution of stress while accounting for the non-linear response in the longitudinal and lateral deformations of the rock material in compression. (d) To provide a means for ascertaining the onset of strength failure in the rock structure and to determine the resulting stress distribution. A simple method of solution which will be used in this paper is that of successive approximations [13]. Since each approximation involves a solution of a linear problem, it is only necessary to provide a means for the computation of the stiffness matrix before each approximation. (a) After the elastic solution for the structure has been reached by the finite-element method, the minimum principal stress in some elements may be tensile and the maximum principal stress compressive. The constitutive equation of the elements in which this occurs

90

G. B A R L A

is that of a transversely isotropic medium. With reference to the coordinate system of the entire structure this equation writes after transformation

{~-} = [AV [c] [a] (,}

(1)

where cos2 fl [A]---- / s i n u/3 / Lsin 13cos fl

sin 2 fl

-- 2 sin flcosfl

]

cos 2fl

2sinfcos3

J

- sin 3 cos 3

(2)

cos z 3 - sin 2 3

for plane stress conditions:

[C] = (1 -- mr2 2)

az

1

0

(3)

0

m(1 - - ncr22 )

for plane strain conditions:

E2 [C] = (1 -k (rx) (1 -- (r~ -- 2no,22)

n(1 --mr2 2) I/n~2(l+ or1)

[

n~2(l + ~ i )

0

(1 -- ol 2)

0 m(1 -k or,)

o

0

] ] (4)

(1 - - tr 1 - - 2ntx22)_]

fl is equal to the angle between the direction of the tensile stress and the horizontal,/71 and E2 are the elastic moduli in the direction of the tensile and compressive stresses respectively. ~x and or2 are the associated Poisson's ratios. G2 is the independent shear modulus, m and n are taken as E~

n : --

G2 m : --

E2'

(5)

E2

and the index T indicates matrix transpose. The stiffness of the rock structure is reformed and a new elastic solution is attained. The process is repeated until the differences in stresses and displacements obtained in successive approximations is negligible. (b) The zone in the rock structure where the process of fracture is initiated is determined by applying the Griffith and modified Griffith criterions of fracture initiation [7]. After solving the structure with tensile--compressive non-linearity for the rock material, the stresses in each element (i) are: {r},=(r~},

(6)

(i= I , . . . , N )

A METHOD FOR THE ANALYSIS OF STRESS IN BRITTLE ROCK

91

where N stands for the n u m b e r of elements in the structure. Therefore, the Griffith and modified Griffith criterions of fracture initiation can be applied as follows: (i) F o r ~(' ] \~1/

i ~< - - 0 . 3 3

(7)

fracture initiates when

(8)

(~)l >/ To (;

=

l ....

, N).

(ii) F o r -- 0"33 ~<

,

~< --

(9)

[1 + 2tL -- 2 V((1 + /z2))]

fracture initiates when

2Co [ ]

(r2) 1~1t ] +[,v/((l -~-/x2))--/z

(~,), >/

0o) (i :

1. . . .

, N).

(iii) F o r

1

(~'2)

[1 + 2~ + 2 V((1 + ~2))] ~< ~ ,

[x/((l q-/x2))- U]

(11)

~< IV((1 + ~2)) + t~]

fracture initiates when

[v'((1 + ~ ) ) - n] Co

(12)

(i = 1. . . .

, N).

(iv) F o r

~-~) /> [V((1 + t~)) - t~] , [V~((1 -I- ~2)) ~_ tt ]

(13)

(i= 1,...,N) no fracture can initiate. Co is the longitudinal stress at fracture initiation under uniaxial loading. To is the tensile strength, and t~ the coefficient of internal friction of the rock material. The finite elements (L) which initiate fracture according to (i) are assumed to obey by the constitutive equation (1). However, in this case the modulus of elasticity (E1)j (j = 1. . . . , L) is set equal to zero. Accordingly, this condition is equivalent to assume that fracture initiation and strength failure in tension occur simultaneously, which is k n o w n to be satisfactory for any practical purpose.

92

G. BARLA

The finite elements, which initiate fracture according to either (ii) or (iii) and for which the minimum principal stress is tensile, are treated as in (a). However, the finite elements (M) which initiate fracture according to (iii) and for which the minimum principal stress is compressive, are treated as an isotropic and linearly elastic medium with the Poisson's ratio (crl)j(j = 1 . . . . . M ) properly increased, in order to account for the already mentioned departure from linearity of the stress-lateral strain curve. For the finite elements which obey (iv), the mechanical properties are naturally left unchanged. The solution is again attained by a process of successive approximations. However, in the present case the solution is accepted when there are no more elements failing according

to (i).

(c) The problem is now that of determining for the rock structure the transition from stable to unstable fracture propagation under compressive load. A criterion exists which allows one to ascertain the onset of unstable fracture propagation [14]. Essentially, fracture propagation becomes unstable when the energy released per unit crack surface attains a known critical value. Experiments have been carried out by BII?NIAWSKI [4] to determine such a critical value also in some rock types. Some attempts have been made by the author to apply this criterion in conjunction with the finite-element method. However, for the purpose of the present investigation, use is made only of a criterion established on purely phenomenological grounds. Therefore, it will be assumed that the onset of unstable fracture propagation takes place whenever

[(~,), - (~,),]/> So JF + AF (~')' + (~2), " 2

(]4)

where So Jr, At, and mr are characteristic constants for the rock material. These constants can be easily determined by plotting on logarithmic scales the results of conventional uniaxial and triaxial compression tests and by fitting the best straight lines to the experimental points by the method of least squares [15]. Clearly, in this case the results are those for the onset of unstable fracture propagation. The finite elements (P) which result to be in a condition of unstable fracture propagation according to the phenomenologicat criterion (14) will be given values of (Ek)l and (ek)~ (k = 1, 2 ; i = 1, P), which account for the non-linear response of both the axial and transversal deformations. A process of successive approximations is used to attain a piece-wise non-linear solution. (d) The criterion used in order to ascertain the strength failure is again established on a purely phenomenological basis. It is assumed for the onset of strength failure

1

[

2

]

05)

where SoJs, As and ms are the same parameters defined in (c)but with reference to the strength failure envelope, as indicated by the subscript S. The finite elements (Q) which fail according to (15) are assumed to have completely lost their load-carrying characteristic and are given zero values of (Ek)~ (k --= 1, 2; i -= 1, Q). Again, the process of successive approximations can be used to attain the corresponding solution. It should be observed at this point that the above assumption does not hold true in real cases. Some rock materials with brittle fracture behavior are known to exhibit, also after

A M E T H O D FOR THE ANALYSIS OF STRESS IN BRITTLE ROCK

93

t REA~ INPUT OATA I KJ=0 ~ JK=0

l

,_ PRINT ELEMENTS MECHANICALPROPERTIES I~ I FORMELEMENT STIFFNESSESI I O~TAINLINEARLY ELASTICSOLUmIONI

I PRINTANOP'OTOISPLACEMENmSANOSTRESSES I I CHECKFOR CONVERGENCE OF NON-LINEAR SOLUTION I

d ~N~TK ~

YES

CHECK FOR ELEMENTS IN TENSION AND ASSIGN MATERIAL PROPERTIES ACCORDING TO (a)

YES

CHECK FOR ELEMENTS IN TENSION -NOT FAILINGAND ASSIGN MATERIAL PROPERTIES ACCORDING TO (a)

NO J ~ TK i ~ NO

t

APPLYCRITERIONS OF FRACTURE INITIATION I PRINT ELEMENTS AND RANGES IN WHICH FRACTURE IS INITIATED

I ASSIGNMATERIALPROPERTIES TO EACHELEMENT I AS IN (b) APPLY CRITERION FOR UNSTABLE FRACTURE PROPAGATION PRINT ELEMENTS IN UNSTABLE FRACTUREPROPAGATION CHANGE MECHANICAL PROPERTIESAS IN (c)

I

APPLYCRITERION FOR STRENGTH FAILURE PRINT ELEMENTS IN STRENGTHFAILURE CHANGE MECHANICAL PROPERTIESAS IN (d)

I ~

~o

NITER ~ = NUMBEROF ITERATIONS NEEDEDFOR CONVERGENCE OF NON-LINEAR SOLUTION NITER 2 = NUMBEROF ITERATIONS NEEDEDFOR COMPLETE SOLUTION

FIG. 2. Simplified flow diagram of the computer program for the non-linear solutions.

v

94

G. BARLA

strength failure, a carrying load characteristic, which depends upon the stiffness of the complete rock mass where the opening is located. A computer program was written in order to perform the above analysis. A simplified flow scheme of this program is shown in Fig. 2. It is apparent that each problem from (a) to (d) can be solved either separately or in a sequence as shown in the flow scheme. In the last case, a solution is attained which accounts for the complete physical behavior of the rock material up to strength failure. Some comments on the convergence of the proposed method of solution are appropriate at this point. One is generally not assured of convergence. However, as shown in the following article, an acceptable solution can be attained after a sufficient number of successive linear analyses. This number is increased as the non-linearity increases. Furthermore, it should be noticed that the final solution is valid only for elastic materials, i.e. it is assumed that the same constitutive equations hold true for either the loading or the unloading processes. NUMERICAL EXAMPLE

In order to illustrate the foregoing theoretical approach, a numerical example for a circular opening located in norite is presented in the following. This particular rock type has been chosen for the analysis as experimental results on the complete physical behavior of norite--up to strength failure--were available [4]. These results for norite tested in compression are reported in Table 1. Then, the diagrams of Figs 3 and 4 are easily derived in order to obtain, as reported in Table 2, the data needed for the mathematical description of the various processes of brittle fracture. It should be observed that for the purpose of the following analysis the coefficients mp and ms are assumed to be equal to unity so that a straight line is fitted to the experimental data as shown in Fig. 4. The material properties TABLE 1. DATAFORNORITETESTEDIN COMPRESSION(BIENIAWSKI[4]) Longitudinal stress Lateral stress oo 38" 2 13" 4 9" 2

Longitudinal stress (psi) at fracture unstable strength initiation fracture failure propagation 12,000 13,350 17,280 21,900

34,000 39,530 54,400 59,000

44,500 69,500 111,800 143,990

TABLE 2. DATA DERIVEDFOR THE MATHEMATICALDESCRIPTION OF THE VARIOUSPROCESSESOF BR1TTLEFRACTURE Fracture initiation Unstable fracture propagation Strength failure

-6831" 63 7108.37

0.750 0" 608 0" 725

SoJl,, SoJs

tL, At, As

psi

A M E T H O D FOR THE ANALYSIS OF STRESS IN BRITTLE ROCK

95

TABLE 3. MATERIAL PROPERTIES ASSUMED FOR THE NORITE IN UNIAXIAL COMPRESSION AND TENSION

Elastic modulus ( X 106 psi) Uniaxial compression up to Fracture initiation Unstable fracture propagation Strength failure Uniaxial tension Fracture initiation, unstable fracture propagation, and strength failure are assumed to occur simultaneously

Poisson's ratio

14" 90 14.90 0" 15

0" 226 0" 298 0" 330

13" 38

0" 226

for the norite in uniaxial compression and tension, throughout the various processes of fracture, are assumed to be given as shown in Table 3. The tensile strength, To is 2525 psi [41. The next step is, as always in the finite-element analysis, to construct an appropriate finite-element model. Due to the geometrical symmetry of the structure and the nature of the stress field here considered (gravity effects are neglected, plane-strain conditions are assumed, and a uniform uniaxial stress field is applied), the model consists of a quarter-plate

i

i 60~ C~

%

j~

50

Strength

failure

-J"

40 [7
3O

\

J

~ a b l e

fracture

\

propagation

\\

,oi

racture initiation \\ \\ \

x\\\

I0

2O

3'0 Normal

/,

40

stress~

\

\\ \\\\

\

1 0

"\

50

\ '\

60

70

103psi

FIG. 3. Fracture initiation, unstable fracture propagation, and strength failure envelopes for norite. Data by BIENIAWSKI [4].

96

G. B A R L A

80

£3.

%

/

70

60 m 50

o~ .cz

40

E E

30

Strengthfailu~

e

20

I0

o

i

,

,o

2o

I

~'o

4o

~'o

I

I

I

6o

7o

8o

9'o

i;o

Meannormalstress, 103psi Fio. 4. Linearized unstable fracture propagation and strength failure envelopes for norite in maximum shear str~s-mean normal stress diagram.

T I /Tv

/ T 2/T v

,J -I

2

3 r/a

Fracture ~ initiation ~ '

Region I " ~ '

Uniaxial stress (~'v) 3000 psi

FIG. 5. Circular opening in a norite rock mass under a uniaxial stress field (~o = 3000 psi)---Fracture zones and principal stresses.

A METHOD FOR THE ANALYSIS OF STRESS IN BRITTLE ROCK

97

'i-l / - r v

o

2

"rz / -rv

I

z

-I "r I / ' r v

T2 /T v

-I-

i

i

I 2

I,

I 3

i

r/o

Fracture initiation

~

Region

I

Region

1-I

Region

1TT

U n i a x i a l s t r e s s ( r v) 5"000 psi

F1G. 6. Circular opening in a norite rock mass under a uniaxial stress field ('~v~ 5000 psi)--Fracture zones and principal stresses. only. Points along the horizontal axis are given zero vertical displacements. Conversely, points along the vertical axis are given zero horizontal displacements. The uniaxial stress field is obtained with a load uniformly distributed on the horizontal outer boundary and by leaving the points along the vertical outer boundary free to displace in any direction. The stress increments in the analysis are set to be equal to 2500 psi, starting with a value of rv = 5000 psi. As the tensile-compressive non-linearity is for norite virtually non-existent, only four iterations were needed in order to obtain the first non-linear solution (i.e. N]TER 1 = 4 in the diagram of Fig. 2). Eight complete cycles were instead made for each stress increment (i.e. NITER 2 = 8 in the diagram of Fig. 2). The numerical results are illustrated in Figs 5-10. Each figure refers to a complete solution. Reported are (in a quarter-plate only) the zones in which the various processes of fracture are expected to occur. Also, the principal stresses, normalized to the applied stress To, are plotted in each figure along the horizontal and vertical axes. Figure 5 shows the solution obtained for To ---- 3000 psi. For this applied stress fracture in region (I) is present at the back and floor of the opening only. The resulting stresses are similar to those derived through the linearly elastic solution. As illustrated in Fig. 6, for an applied stress % = 5000 psi the fracture zones at the back and floor are extended and fracture initiates, in region (II), at four points in the structure, ROCK

911--O

98

G. BARLA

T l /'r

v

3

-I

'f TI / r v

"rz/'r v

4-

I

I 2

t

I 3

[

--

r/a

Fracture initiation

lfiiiiiii

Region .T.

~

Region EZ

U n i o x i a l stress (r v)

7500

psi

Region rrr FIG. 7. Circular opening in a norite rock mass under a uniaxial stress field (To = 7500 psi)--Fracture zones

and principal stresses. and in region (III), at the ribs. A change in the distribution of stress along the vertical axis occurs. The stresses along the horizontal axis, however, are not affected. If the applied stress ~'ois further increased, the three zones in which fracture occurs extend. Figure 7 refers to ro = 7500 psi. It shows, when compared with Fig. 6, that fracture in region (I) propagates along the vertical axis and from the opening outwards. Notable zones in which fracture is initiated in either region (II) or (III) are now present. While the distribution of stress along the vertical axis is again changed, the stresses along the horizontal are practically equal to those obtained with the linearly elastic solution. Figures 8 and 9 show that the fracture in region (I) stabilizes, when the applied stress is increased. However, fracture in region (II) moves toward the vertical axis and fracture in region (III) extends to a major portion of the structure. The distribution of stress is in both cases equal to that of the previous figure. A considerable change in the fracture pattern occurs for To = 15,000 psi (Fig. 10). The complete solution obtained in this case shows two zones in which the processes of unstable fracture propagation and strength failure occur in the proximity of the opening ribs. The distribution of stress along the horizontal axis is changed considerably and a transfer of stress from the destroyed zone, in the near vicinity of the opening, to the rock under fracture initiation takes place. The stresses along the vertical axis remain the same. A further increment of 2500 psi in the applied stress leads to an unstable solution. Therefore, it is concluded that, according to the present method of analysis, strength failure in a

A METHOD FOR THE ANALYSIS OF STRESS IN BRITTLE ROCK

99

norite rock mass containing a circular opening would occur, under uniaxial stress field, for % = 15,000 psi, whereas the compressive strength for the norite is 44,500 psi. Also, a further increase in the applied stress of 2500 psi would result in the complete failure of the rock structure.

T2/rv I

-I

O

,.-r

~

_

i

rl /rv "Z" 2 / , r

d-

I I

2

v

i 3

r/a

Region Fracture initiation

I

[TTTT'~ Region

]].

Region

£ff

Uniaxial stress(r~) I 0 , 0 0 0 psi

FIG. 8. Circular opening in a norite rock mass under a uniaxial stress field (~-~= 10,000 psi)--Fracture zones and principal stresses.

CONCLUDING REMARKS The theoretical method proposed in this paper allows one to effect a stability analysis of a rock structure. The complete physical behavior of a rock material which exhibits brittle fracture is considered. The zones in the rock structure which undergo the various processes of fracture (i.e. fracture initiation, unstable fracture propagation and strength failure) can be determined and the distribution of stress computed. The Griffith and modified Griffith criterions are used for predicting fracture initiation. Two criterions, formulated on a purely p h e n o m e n o logical basis, are introduced in order to determine the onset of unstable fracture propagation and strength failure. It is assumed that fracture initiation and strength failure in tension occur simultaneously.

I00

G. B A R L A

The method of solution is based upon the application of the finite-element analysis and a process of successive approximations. The applied stress is increased by steps until a complete failure o f the structure is reached. It is implied that the strength failure of the rock structure is achieved whenever it is impossible to make the procedure convergent,

L.

T2/T v

-I

0

TI / T v T 2 /TV

+

--.r-.-.-.3

2 r/o

iEE~Lq] Region

Fraction [TITT~] initiation I-'----]

I

Region 1"[ Region

U n i o x i a l stress ( r v) 12,5 O0 psi

TI-[

FIG. 9. Circular opening in a norite rock mass under a uniaxial stress field (zv = 12,500psi)--Fracture zones and principal stresses.

A numerical example for a norite rock mass containing a circular opening shows that a careful description of the fracture processes which are expected to occur in the rock structure, under increasing compressive load, can be effected. In particular, it has been shown that under a uniaxial stress field the fracture would initiate at the back and floor o f the opening and it would propagate along the vertical axis, from the opening outwards. As the applied stress is increased, a region in the near vicinity of the opening ribs undergoes, in sequence, the processes of fracture initiation, unstable fracture propagation and strength failure. If the resulting distribution of stress is compared with the one obtained by the linearly elastic solution, it is seen that, after the initiation of fracture in the back and floor regions, the stresses along the vertical axis are changed considerably. Also, the initiation of fracture in other regions of the structure does not modify the stresses along the horizontal axis. The same stresses are greatly affected after the onset of unstable fracture p ~ p a g a t i o n and strength failure at the ribs.

A METHOD FOR THE ANALYSIS OF STRESS IN BRITTLE ROCK

~i ~L

101

.:~.

]'-2 / "¢v

I -I

"1"I / - r v .,.....__..._ "1"2 / - r v

+

o

I

3

2

r/a

Fracture initiotion

I-~

Region

I

Region

T[

U n i a x i o l stress ('iv) 15, 0 0 0 psi

Region ]I][ Unstoble fracture propagation Strength failure

FIG. 10. Circular opening in a norite rock mass under a uniaxial stress field (~'v = 15,000 psi)--Fracture zones and principal stresses.

REFERENCES I. BLAKE W. Application of the finite-element method in solving boundary values problems in rock mechanics. Int. J. Rock Mech. Min. Sci. 3, 169-180 (1966). 2. BARLA G. The Distribution of Stress around Underground Openings--Effects of some Geologic and Mechanical Features of the Rock Mass, Proe. il primo convegno lnternazionale sui Problemi Tecnici nella Costruzione di Gallerie, Torino (1969). 3. ZmNKIEWICZO, C., VALLIAPPANS. and KINO I. P. Stress analysis of rock as a 'no tension' material. G~oteehnique 18, 56-66 (1968). 4. BIENIAWSKIZ. T. Mechanism of Brittle Fracture of Rock, Report CSIR (S. Aft) No. 580 (1967). 5. GRIFFITHA. A. Theory of Rupture, Proceedings of the International Congress of Applied Mechanics, pp. 55-63, J. Waltman Jr, Delft (1925). 6. McCLl~rrocK F. A. and WALSH J. B. Friction on Griffith Cracks in Rocks under Pressure, Proceedings of the Fourth U.S. National Congress of Applied Mechanics, pp. 1015-1021 (1963). 7. HOEKE. Rock Fracture around Mining Excavations, Proceedings of the Fourth International Conference on Strata Control in Rock Mechanics, pp. 334~348, New York (1964). 8. BIENIAWSKIZ. T. and VAN TONDER C. P. G. A photoelastic model study of stress distribution and rock fracture around mining excavations. Expl Mech. 9, 75-81 (1969). 9. BIENIAWSKIZ. Y. Mechanism of Rock Fracture in Compression, Report CSIR (S. Afr.) No. 459 (1966). 10. BRACEW. F., PAULDINGB. W. and SCHOLZ C. Dilatancy in the fracture of crystalline rocks. J. geophys. Res. 71, 3939-3953 (1966). 11. BARLA G. Some Constitutive Equations for Rock Materials, Proceedings of the Eleventh Symposium on Rock Mechanics, Berkeley (1969).

102

G. BARLA

12. Z~ENK~wI~z ~. C. and CHEUN~Y . K . The Finite E~ement Meth~d in Structural and C~ntmuum`~echanics~ McGraw-Hill, London (1967). 13. WILSONE. L. Finite Element Analysis of Two-dimensional Structure, Doctoral Dissertation, University of California, Berkeley (1963). 14, IRWIN G. R. Fracture Mechanics, in Structural Mechanics (J. N. Goodier and N. J. Hoff, Eds) pp. 557-592, Pergamon Press (1960). 15. HOEKE. Brittle Fracture of Rock, in Rock Mechanics in Engineering Practice, pp. 99-124, Wiley, New York (1968).